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Is there an Einstein metric on a compact nilmanifold (compact quotient of a non-abelian nilpotent Lie group) in dimension > 3 ?

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    $\begingroup$ At least, non-abelian nilpotent Lie groups do not admit left-invariant Einstein metrics (Milnor). $\endgroup$
    – Dietrich Burde
    Commented Aug 23, 2017 at 19:37
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    $\begingroup$ Thanks AP :-) These compact non-abelian nilmanifolds do not admit any metric with non-negative Ricci curvature, so the Einstein metric, if it exists, must have negative Einstein constant. Even for the torus it's still not known (as far as I am aware of) whether there is an Einstein metric with negative constant. $\endgroup$
    – Min-Oo
    Commented Aug 24, 2017 at 4:54

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